Estimation of Conditional Mean Operator under the Bandable Covariance Structure

TL;DR

This paper develops minimax optimal estimators for the regression coefficients in high-dimensional multivariate linear models with bandable covariance matrices, improving upon existing methods through novel blockwise tapering techniques and Bayesian procedures.

Contribution

It introduces the blockwise tapering estimator and a Bayesian post-processed posterior that achieve minimax optimal convergence rates for the regression coefficient under bandable covariance structures.

Findings

Proposed estimators outperform existing methods in simulations.

Blockwise tapering estimator achieves minimax optimal convergence.

Bayesian procedure also attains minimax optimality.

Abstract

We consider high-dimensional multivariate linear regression models, where the joint distribution of covariates and response variables is a multivariate normal distribution with a bandable covariance matrix. The main goal of this paper is to estimate the regression coefficient matrix, which is a function of the bandable covariance matrix. Although the tapering estimator of covariance has the minimax optimal convergence rate for the class of bandable covariances, we show that it has a sub-optimal convergence rate for the regression coefficient; that is, a minimax estimator for the class of bandable covariances may not be a minimax estimator for its functionals. We propose the blockwise tapering estimator of the regression coefficient, which has the minimax optimal convergence rate for the regression coefficient under the bandable covariance assumption. We also propose a Bayesian procedure called the blockwise tapering post-processed posterior of the regression coefficient and show that the proposed Bayesian procedure has the minimax optimal convergence rate for the regression coefficient under the bandable covariance assumption. We show that the proposed methods outperform the existing methods via numerical studies.